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A Guide to Game Theory and Negotiations

2009 November 29

Everybody negotiates.  Some people like it, others hate it but all agree that the ability to negotiate well is a valuable business skill and therefore it’s something worth doing well.

To be successful in negotiations, you have to be tough, but it also helps to have a strategy.  Fortunately, Game Theory provides us with insights that can lead to practical results.

The Simplest Game: Two Person with a Fixed Pie

Usually when we think of negotiations, we imagine two people facing off against each other.  Whatever one wins, the other loses.  These were the first type of games that Game Theory attempted to solve, so it’s a good place to get started.

Look at the following example where Neil owes Bob some money and they have to agree on how much.  They each have two strategies available to them, but the outcome is not just dependent on the strategy chosen, but also on what their opponent does. So while Neil would obviously choose to pay the minimum (4) and Bob would like to have the maximum (6) the opposing move needs to be taken into account before an optimal strategy can be selected.

Neil Pays Bob (2 Strategies)

Neil Pays Bob: 2 Strategies

This is about the simplest game imaginable, so you don’t have to be a game theorist to figure out that the payoff will most likely be 5, but it illustrates a very good rule called the minimax theorem.  The concept, which was originated by the legendary John von Neumann, simply says that you should pick the strategy where the maximum advantage of your opponent is minimized.

In the above example, Neil would pick strategy #2 to avoid the possibility of having to pay 6 while Bob would pick strategy #1 to avoid only getting paid 4.

Minimax is a simple but useful rule that can be applied to everyday situations.  However, it’s problematic partly because it assumes perfect information.  In the real world, things are much more confusing, not least because our negotiating opponent  is rarely limited to two strategies.

Let’s see what happens if we add additional strategic possibilities.

Neil Pays Bob: 4 Strategies

Neil Pays Bob: 4 Strategies

Mathematically, the game is still solvable, but you can see that it gets much more complicated just by adding a few extra strategic choices.  Therefore, it is generally an advantage to have less options for yourself and more options for your opponent (although a bit counter-intuitive).

Game theorists achieve this mathematically by determining which strategies are numerically dominant and using randomly mixed strategies.  In real life, you just need to have a clear idea of what you want and eliminate inferior options.  Your advantage can sometimes be increased if you don’t announce your preferences to your opponent so that he still has to consider variables which you don’t.

Reducing variables for yourself is almost always a good idea. Ironically, someone with a gun to his head is in a very strong negotiating position (with people other than the gunman of course) because he has few choices .  In the world of business, we usually refer to a boss or shareholders in order to show that our options are limited, but the principle is the same.

In reality negotiations have many complications.  For instance, it makes a big difference if one player chooses first, whether they play one time or if it’s an ongoing game, whether there are side deals, etc.

One particularly interesting real world difficulty that we’ll discuss next is that we often find that some agreements seem to be in place even before we start negotiating!

Bargaining without Bargaining

When you negotiate internationally, you often hear the phrase “everybody knows that it works this way.”  This can be very frustrating; not least because it so often follows you saying “that’s the craziest thing I’ve ever heard!”

This phenomenon is a result of “tacit bargaining,” a term first coined by Thomas Schelling (he later won the Nobel Prize for his work).  It’s more common than you would think.  Consider the following experimental results cited by Schelling:

  • When asked to pick any number, 40% chose the number 1.
  • The overwhelming majority of people chose heads over tails on a coin toss
  • When asked to pick any amount of money almost all people chose a figure divisible by 10
  • When people where told that they had to meet someone else – but had to guess the time – almost all chose noon.
  • An overwhelming majority of people would rather pay dues than taxes

We often concede to convention without even knowing it (although it becomes much more apparent when we encounter customs we’re not used to).  It is natural for us to want to abide by notions of fairness and precedence.  Nobody likes to be seen as rocking the boat.

Therefore, it is generally an advantage to make the first move in a negotiation, even though most people are reticent to do so.  Making the first proposal gives you the opportunity to frame the negotiation and establish precedence.

Another insight gained from research into tacit bargaining is that often an outsider can be more effective.  That’s why when you start getting comfortable with a car salesman you inevitably end up in the manager’s office to close the deal.

Making Commitments: Promises and Threats

Zero sum negotiations are actually quite rare.  We usually enter negotiations because we expect the results to enlarge the total pie and  make both sides better off.  In effect, we prevail with a mixture of cooperation and competition andwe will have to make commitments in the form of promises and threats.

Let’s look at some examples.  Because these are non zero sum examples, we show payoffs to both players .  In each box, Bob gets the payoff colored in red while Neil gets the payoff colored in green.

Bob wins by threatening Strategy #2

Bob wins by threatening Strategy #2

In the above example, it seems at first that Neil has a much better hand.  He can gain from either of his strategies while only one of Bob’s strategies results in a payoff.

However, if Bob threatens to pick strategy #2 if Neil picks #1, he can come out ahead as long as the threat is credible. Since Neil will lose more than he will, it’s a very viable course of action.

In the next example, there is only one scenario where both players can gain.  For Bob to gain something, he has to promise Neil that he won’t be greedy.

Non Zero Sum Promises

Bob must promise Strategy #2

While it might seem that Bob’s better strategy is #1, the only way he will be actually able to gain anything is by promising Neil that he will pick strategy #2.  If he doesn’t, Neil will pick his strategy #1 in order to avoid a loss.

In this next example, a promise of a side payment turns a very bad position into a a big winner.  Bob wins by promising a side payment of $2

Bob wins by promising a side payment

Bob wins by promising a side payment

By offering the side payment, Neil’s payoff for choosing his strategy #2 is superior to his strategy #1.  Morover, he can feel confident that Bob will choose strategy #1 because that is now Bob’s dominant strategy.  So Neil will gain 3 and Bob will gain 8.

While we all value freedom, in negotiations the commitments we make can be extremely valuable, if they’re credible. Since they limit our options, it often takes some effort to get our negotiating partner to believe in us.  Contrary to the stereotype, the last thing a good negotiator wants is to be seen as is “slick.”

Prisoner’s Dilemma

The most famous and interesting example in Game Theory is the Prisoners Dilemma.

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The basic narrative should be familiar to anyone who’s watched a cop show.  Two suspects are brought in and interrogated separately.

If they effectively cooperate with each other they will get off on a lesser charge and serve only two years.  If one defects, he will only get one year in prison while his friend gets a harsh five year sentence.  If they both defect, they will each get three years.

(note: There are many variations of the Prisoner’s Dilemma that use different values).

While they would both obviously benefit from keeping their mouths shut, individually the dominant strategy is to talk. By confessing, each will be better off no matter what the other one does and by holding out each will be worse off.

This is an example of  the Nash equilibrium made famous in the movie A Beautiful Mind and shows how sometimes a stable solution can also be a bad one. In effect, everybody can agree to be worse off.

Three quick points about the Prisoner’s Dilemma before we move on:

  • Trust: Building credibility is essential to avoiding defection
  • Tit for Tat: Experimentally, the most successful solution has been shown to be “Tit for Tat” first formulated by Anatol Rapoport, where you reciprocate your opponent’s last move.  Cooperation brings cooperation and defection brings defection.
  • Continuity: If the prisoners dilemma is open ended, people have more reason to cooperate.  However, if the engagement is limited they will be more likely to defect.  The reason why is called a backward induction paradox.  If there are 100 games, then it makes sense to cheat on the last one.  But if each player knows that,  it makes sense for him to get ahead by cheating on the 99th.  The same logic would hold for the 98th and so on all the way back to the first game.

Bigger Games

Of course, many games involve more than two people and most negotiations take place in a larger context.  A salary negotiation will ultimately affect other salaries in the same company, a price negotiation will have ripple effects for an industry, etc.

Games with more than two people are called N-person games.  Understandably, they are much more complex than two person games and we can’t do them justice here.  However, I think it’s worth it to go over a few basic points:

  • All N-person games are defined as zero sum.
  • Most solutions are based on voting models and coalition building.
  • Your power is derived from the probability of casting the 51st vote out of 100.
  • Size isn’t always an advantage.  In a company that has a share structure of 40-40-20, all shareholders would have equal power.  If the two bigger players are antagonists, the smallest player can actually run the show!

Practical Applications for Real World Negotiations

While it’s important to remember that Game Theory consists of sanitized models with often unrealistic assumptions, we can derive some useful principles that can improve results in real world negotiations:

  • Minimize Risk: As the minimax theorum shows, all things being equal it is usually a good strategy to minimize your opponents maximum payout, even if that means that you are guaranteeing him a higher minimum.
  • Make the first offer: Unless you are at a demonstrable informational disadvantage, it is better to go first.  This allows you to frame the negotiation to your advantage.  Although it’s against most people’s natural inclination, the evidence is pretty strong here.
  • Beware of Precedence: While it’s difficult to go against convention, there is no reason to simply accept a situation where you are at a disadvantage.
  • Trust: The ability to make commitments such as promises and threats can be extremely valuable.  If you lack credibility, you lose that capacity.  Sometimes it’s better to incur a cost rather than lose stature.
  • Continuity: Open ended commitments encourage reliability while short term relationships abet duplicity.  That’s why all tourists are suckers.
  • Reciprocity: Rapoport’s “Tit for Tat” strategy and subsequent evidence shows that you can expect to get what you give.  Moreover, while there is no reason to set out on a nasty path, sometimes that’s where you’ll have to end up.  As the Al Capone character in The Untouchables saidYou get more with a kind word…and a gun, than you do with just a kind word.

I hope this has been helpful.  Please let me know your comments.

– Greg


16 Responses leave one →
  1. November 30, 2009

    Says Dr. Lawrence Cohen:

    “Wherever complex cooperation has been able to evolve, it is because cooperators have been able to capture the effects on others of their actions. As a result, self-interest has driven them to treat the other as self. A strong example that may not be immediately obvious is an economic market. The market will reward an individual who develops a new product that benefits others by satisfying their needs better than existing products. Selling the product enables the individual to capture the beneficial effects that the product has on others. In the limited areas where economic markets work effectively, individuals benefit in this way from actions that benefit others. Where the market enables individuals to capture the benefits of their effects on others, the market aligns the interests of individuals with the interests of others.

    In contrast, where the barrier to the evolution of cooperation has not yet been overcome, cooperators do not capture the beneficial effects on others of their actions. For example, in our economic system, a person will not capture the benefits of providing food to people who are starving in an African famine. It would be far more profitable to invent an improved mousetrap. Or a new weapon of mass destruction. But when the barriers to cooperation in human society are overcome, and when all individuals capture in full their beneficial effects on others, the provision of food to the starving will be a lucrative way to make a living.”

    [Reply]

  2. November 30, 2009

    Cosmides:

    “The game theoretic models for the evolution of cooperation that could be reasonably applied … require the existence of some mechanism for detecting cheaters or otherwise excluding them from the benefits of cooperation. This is because the capacity to engage in social exchange could not have evolved in the first place unless the individuals involved could avoid being continually exploited by cheaters. But most models do not require the existence of a mechanism for detecting “altruists” individuals who follow the strategy of paying the required cost (thereby benefiting the other party), but not accepting from the other party the benefit to which this act entitles them. Indeed because individuals who were consistently altruistic would incur costs but receive no compensating benefits, under most plausible scenarios they would be selected out. From the perspective of deontic thinking, we would expect no difference in facilitation.”

    [Reply]

  3. December 2, 2009

    Hi Greg.

    Another good article. Game theory is fascinating stuff, but just understanding it can be challenging sometimes. You were able to present the concepts in a relatively easy to understand way.

    [Reply]

    Greg Reply:

    Steve,

    Thanks. That’s very kind of you.

    – Greg

    [Reply]

  4. Joan permalink
    December 19, 2009

    Hi Greg,
    I was interested in your article but could not follow it in terms of real-life conversations people can have with their bosses. I would like to hear negotiating tips and what has worked for you in the past. The game theory is not personal and I like personal….

    [Reply]

    Greg Reply:

    Joan,

    Well, if I could boil it down to one point (which I can’t) I would say that the one thing to get right is credibility. Most os negotiation tactics will only be effective if the other side believes us.

    I hope that helps.

    Have a nice holiday.

    – Greg

    [Reply]

  5. December 23, 2009

    Hi Greg,

    Another lucid and thought provoking item. I’m about to embark on a complicated negotiating process with a possible partner although we have previously been competitors. This has helped me simplify and clarify my approach.

    many thanks

    Paul

    [Reply]

    Greg Reply:

    Paul,

    Thanks. I’m glad it was helpful.

    Happy Holidays!

    – Greg

    [Reply]

  6. Peter permalink
    March 13, 2010

    Greg,

    Good article. I have always wondered about practical use of Game Theory.

    However, some Case Studies may help.

    regards,
    Peter

    [Reply]

  7. aron kenyon permalink
    June 21, 2011

    Gregg is was very helpful as i copy and pasted it for a piece if work

    [Reply]

    Greg Reply:

    Glad to hear it!

    – Greg

    [Reply]

  8. August 30, 2012

    I am working on developing a video game for people to practice salary negotiations. I’d love you input! http://www.salarynegotiators.com/become-an-expert-salary-negotiator-video-game-style/

    [Reply]

    Greg Reply:

    Sure. Let me know what I can do. You’ll find my LinkedIn profile on the “about” page.

    – Greg

    [Reply]

  9. Moe permalink
    December 7, 2012

    good article!
    Tit-for-tat is a good strategy in a repeated version of prisoner’s dilemma.

    [Reply]

    Greg Reply:

    Yes, it is!

    – Greg

    [Reply]

  10. August 11, 2013

    Great article about the art of negotiation. To negotiate or not to negotiate – that is the question.
    Love the game theory.

    [Reply]

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